![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > recgt1i | Structured version Visualization version GIF version |
Description: The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.) |
Ref | Expression |
---|---|
recgt1i | ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (0 < (1 / 𝐴) ∧ (1 / 𝐴) < 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0lt1 11151 | . . . . 5 ⊢ 0 < 1 | |
2 | 0re 10632 | . . . . . 6 ⊢ 0 ∈ ℝ | |
3 | 1re 10630 | . . . . . 6 ⊢ 1 ∈ ℝ | |
4 | lttr 10706 | . . . . . 6 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < 1 ∧ 1 < 𝐴) → 0 < 𝐴)) | |
5 | 2, 3, 4 | mp3an12 1448 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((0 < 1 ∧ 1 < 𝐴) → 0 < 𝐴)) |
6 | 1, 5 | mpani 695 | . . . 4 ⊢ (𝐴 ∈ ℝ → (1 < 𝐴 → 0 < 𝐴)) |
7 | 6 | imdistani 572 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (𝐴 ∈ ℝ ∧ 0 < 𝐴)) |
8 | recgt0 11475 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < (1 / 𝐴)) | |
9 | 7, 8 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → 0 < (1 / 𝐴)) |
10 | recgt1 11525 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐴 ↔ (1 / 𝐴) < 1)) | |
11 | 10 | biimpa 480 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ 1 < 𝐴) → (1 / 𝐴) < 1) |
12 | 7, 11 | sylancom 591 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (1 / 𝐴) < 1) |
13 | 9, 12 | jca 515 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (0 < (1 / 𝐴) ∧ (1 / 𝐴) < 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 < clt 10664 / cdiv 11286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 |
This theorem is referenced by: recnz 12045 xov1plusxeqvd 12876 log2tlbnd 25531 padicabvf 26215 rtprmirr 39502 stoweidlem34 42676 eenglngeehlnmlem2 45152 |
Copyright terms: Public domain | W3C validator |